A model for predicting pellet-cladding interaction-induced fuel rod failure

EL SE Vl ER

Nuclear Engineering and Design 156 (1995) 393 399

Nuclear Engineering and Design

A model for predicting pellet-cladding interaction-induced fuel rod failure Lars Olof Jernkvist Asea Brown Boveri Atom, S-721 63 Vgisterds, Sweden

Abstract A model for predicting pellet-cladding mechanical-interaction-induced fuel rod failure is presented. Cladding failure is predicted by explicitly modelling the formation and propagation of radial cladding cracks by the use of non-linear fracture mechanics concepts in a finite element computational framework. The failure model is intended for implementation in finite element fuel performance codes in which local pellet-clad interaction is modelled. Crack initiation is supposed to take place at pre-existing cladding flaws, the size of which is estimated by simple probabilistic concepts, and the subsequent crack propagation is assumed to be due to either iodine-induced stress corrosion cracking or ductile fracture. The novelty of the outlined approach is that the development of cladding cracks which may ultimately lead to fuel rod failure can be treated as a dynamic and time-dependent process. The influence of complex or cyclic loading, ramp rates and material creep on the failure mechanism can thereby be investigated. The presented failure model has been incorporated in the ABB Atom STAV-Ttransient fuel performance code. Numerical results from some applications of the code are used to illustrate the usefulness of the model.

I. Introduction Fuel rod failure due to mechanical and chemical interaction between fuel pellets and cladding (PCI) has been the topic of many investigations during the last two decades. The effort spent on elucidating the initiation and propagation of cladding cracks under the combined effects of mechanical straining and exposure to corrosive fission products bears witness to the complexity of this failure process. Three properties of in-reactor PCI-induced fuel rod failures which considerably complicate any approach to numerically analyse the failure process can be identified.

(1) Cladding stresses and strains are extremely localized owing to both radial cracking and axial hour-glassing of the fuel pellet. The vast majority of PCI-induced cracks are usually found near pellet radial cracks and/or at pellet-pellet interfaces. This pronounced spatial localization of damage can be explained by the concentration of stresses and the abundance of released fission products in these areas. (2) Fuel rod failures due to PCI usually occur following sudden increases in power. This may be due to both intense fission gas release and insufficient relaxation of cladding stresses during rapid power excursions.

0029-5493/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved S S D I 0029-5493(94)00961-9

394

L.O. JernkvLs't / Nueh, ar Engineering and Design 156 (1995) 393 399

(3) In-reactor PCI-induced fuel rod failures usually show strong variability. Out of maybe a thousand similar fuel rods subjected to the same loading, only one or two will fail. This random occurrence of in-reactor failures is in glaring contrast with laboratory failure tests, generally exhibiting a fair reproducibility. These three features have been borne in mind when developing the failure model. The localization of failure in both space and time is tackled by detailed finite element modelling of the pelletclad interaction in a transient fuel performance code, whereas the stochastic nature of in-reactor failures is approached by simple concepts from probabilistic fracture mechanics. In Sections 2 and 3 the models for initiation and propagation of cladding cracks are presented. Section 4 is devoted to the calculation of crucial crack tip parameters. Finally, the implementation and application of the failure model are briefly described in Sections 5 and 6.

suited for studies of in-reactor failures where large quantities of cladding are subjected to similar loading. Following the above assumption, a condition for initiation of in-reactor PCI cracks could be based on the probability of finding surface flaws with lengths exceeding a critical threshold dictated by the current mechanical loading. This probability may be quantified using Weibull's theory for probabilistic fracture mechanics in brittle materials (Weibull, 1951). According to this theory, the probability F of finding a body with area A having a strength less than a is F(a) = 1 - e x p [ - AJ(o-)]

(1)

where J ( a ) is the probability of failure per unit area. Usually f can be written as

(2)

--

where Ao, ao and m are constants specific to the 2.

Crack

initiation

The mechanisms governing the formation of iodine-induced stress corrosion cracks in zircaloy are not yet fully understood, although much experimental work has been done to clarify the matter (Brunisholz, 1987; Cox, 1987). The experiments, usually performed by pressurizing short pieces of zircaloy tubes in the presence of iodine, suggest an initial slow intergranular mode of propagation, followed by a sudden transition to rapid transgranular crack growth as soon as a critical stress intensity is reached at the crack tip. In the failure model presented here, the first stage of slow intergranular crack growth is disregarded. Instead, cladding cracks are assumed to initiate at pre-existing internal flaws sufficiently large to generate the stress intensity needed for immediate onset of transgranular crack growth. This approach is vindicated only if there is a non-vanishing probability of finding a large enough surface defect in the material. Since this probability obviously increases with the volume of material under consideration, the approach is

material. Applying these relations to PCI-induced cladding failure, we first have to assume that crack propagation occurs when the stress intensity J at a surface flaw of length a reaches a critical threshold value J ~ ka = Jsc~-

(3)

In Eq. (3) J has been assumed proportional to the crack length, which is true for brittle materials. Js~ is the threshold stress intensity for transgranular crack growth. Assembling Eqs. (1)-(3), we arrive at a distribution function for the surface defects:

[

F(a) = 1 - exp -~-~

(4)

Here F expresses the probability of finding a surface flaw longer than a in a body with area A. It should be noticed that the form of F has been deduced merely by assuming that zircaloy subjected to PCI behaves as a brittle material highly sensitive to surface flaws. It has been shown by Miller et al. (1981), however, that the actual distribution of surface defects in zircaloy is well described by Eq. (4), where A' o, ao and m' may be established

L.O. Jernkvist / Nuclear Engineering and Design 156 (1995) 393-395

by investigating the distribution of flaws in a moderate number of samples of the clad material. In the failure model, Eq. (4) is used to determine the longest pre-existing surface flaw which, with any extent of confidence, is expected in the cladding area under study. This probabilistic treatment of the crack initiation process is justified by the stochastic nature of in-reactor PCI failures.

3. Crack propagation In our failure model, cladding cracks are assumed to propagate owing either to iodine-induced transgranular stress corrosion cracking or ductile fracture. Iodine-induced stress corrosion (I-SCC) is considered as the leading mechanism for propagation under moderate loading provided that there is a sufficient quantity of iodine available in the neartip region. Ductile fracture may occur, however, under intense mechanical straining of the cladding or when I-SCC is hindered by either a depleted chemical environment or a pronounced radial texture of the cladding material (Ryu, 1988; Schuster, 1992). Another assumption that we have made is that the two modes of crack growth at any instant can be treated as non-interactive. This is generally not true, but in the vast majority of cases PCI-induced cracks will propagate mainly via brittle transgranular stress corrosion cracking until the remaining ligament fails, in some cases under unstable ductile fracture. In such a case it is justified to believe that the incremental crack growth under a very short time is due solely to either I-SCC or ductile fracture. The failure mechanism yielding the dominant contribution to crack growth under this time step is supposed to be the current mode of propagation. In either of these failure mechanisms the crack propagation velocity at any instant is assumed to be controlled by the current conditions at the crack tip, with particular respect to stress intensity, temperature and iodine concentration. These crack tip parameters, supplied to the failure model from the finite element host code,

395

are used to determine the current crack growth velocity from material correlations described in the following sections. The crack propagation velocity is finally returned to the host code, where the finite element model of the cladding geometry is modified to accommodate the crack extension during the current time step. This explicit modelling of crack propagation is performed using a finite element node release technique (Shih, 1979). Both stress intensity and availability of iodine at the crack tip are time-dependent parameters that have to be repeatedly determined during a prescribed power history. This time dependence is partly due to changes in heat generation rate, but the major contribution in many cases arises from the crack growth itself. By the explicit finite element modelling of the propagating crack, these effects can be taken into account. 3.1. Propagation due to I-SCC The crack propagation veloclity da/dt under iodine-induced transgranular stress corrosion cracking can be expressed by a correlation of the form __d°= {0 (J---~c~)J ~ (RQT)ifJJscc where C (m s- ~) is a constant, n is a non-dimensional constant, R (J mol -~ K -~) is the molar gas constant, Q (J m o l - ~) is the activation energy, Js¢c (N m-~) is the I-SCC threshold stress intensity, F is a non-dimensional function of the iodine concentration, J (N m-~) is the stress intensity and T (K) is the temperature. The parameters Q and C are dependent on the material under consideration, whereas n is approximately the same for both Zircaloy-2, Zircaloy-4 and unalloyed zirconium and can be found in the range 1.35-1.50 (Nagai, 1985). The threshold stress intensity for transgranular stress corrosion, Jsc~, is dependent mainly on the texture of the material but is also influenced by temperature and irradiation (Brunisholz, 1987; Schuster, 1993). The threshold value for Zircaloy4 under normal operating conditions can be found

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L.O. Jernkvist / Nuclear Engineering and Design 156 (1995) 393-399

1,0

0,8 7

~_0,6

!

0,4

30-

0,2

0,0

........

10-5

i ........ 10 .4

Iodine

13 10

........

i 10 .2

Concentration

.......

~ ...... 10 "1

[ m o l e / m 9]

Fig. 1. The non-dimensional function F(I2) from Eq. (5).

in the range 2 5 0 - 1 0 0 0 N m -~. The non-dimensional function F(I2) is shown in Fig. 1 and the crack propagation velocity in unirradiated Zircaloy-4 as a function of stress intensity at two different temperatures is shown in Fig. 2. 3.2. Propagation due to ductile fracture

At extremely high stress intensities the crack growth is assumed to be controlled by the J 1,00

~0

0

i

i

i

i

2

4

6

8

10

Ductile C r a c k E x t e n s i o n [ ~ m ] Fig. 3. An idealized JR curve of zircaloy. Jw is the material fracture toughness, which is dependent on temperature and irradiation. resistence (JR) curve of the cladding material (Shih, 1979). Since no Jg measurements have been reported on irradiated ziracaloy, the idealized JR curve shown in Fig. 3 is used to determine the incremental crack growth under ductile fracture. Given the current stress intensity J(a) and the accumulated crack growth due to ductile fracture, aa, the incremental crack growth Aa can be found by imposing the crack propagation to follow the JR curve:

/ " Aa = 0

Unirradiated Zr-4 at saturated

.Oon /

~-~ 0,80

iodi~

j

con-

j

r

,

r

if J(a) < JR(aa)

(6)

Otherwise Aa is found from the JR curve via the condition JR(aj + Aa) = J(a)

, ~ 0,60

f /

o, o

- T- c:

4.

Calculation

of crack

tip parameters

4.1. Stress intensity

0,00

400

, 600

•

, 800

Stress

'

, I000

Intensity

•

, • , • 1200 1400 1600

The stress intensity at cladding cracks is calculated using the J integral first introduced by Rice (1968):

[ N/m ]

Fig. 2. Crack propagation velocity in unirradiated Zircaloy-4 at saturated iodine concentration according to Eq. (5).

J= f

W dy - ao.nj au~ 0---£ds

(7)

L.O. Jernkvist / Nuclear Engineering and Design 156 (1995) 393-397

Here W denotes the deformation work per unit volume of material, F is any closed path around the crack tip, a o n j is the outward surface traction and u, is the displacement along this path. J can be numerically evaluated according to its definition in Eq. (7) by using the near-tip stress and displacement fields acquired from the finite element host code. J must be re-evaluated whenever the load or crack geometry changes, in order to reflect the curi-ent stress intensity at the crack tip. 4.2. I o d i n e

concentration

The total inventory of iodine available in the fuel rod void volume may be divided into two parts: (1) stable or extremely long-lived isotopes (127I, 129I); (2) short-lived isotopes (chiefly 131I). At rod burn-ups above 30 MWd kgU 1 the fuel rod inventory is generally dominated by 129I, but at lower burn-ups the short-lived isotopes may yield a significant contribution to the overall iodine inventory. The release and build-up of stable isotopes may be modelled in much the same manner as the release of inert gases (Forsberg, 1985). The release of short-lived isotopes is handled by a simple model described below. Under steady state a short-lived isotope in the fuel rod void volume will attain an equilibrium concentration which is determined by the isotope's fission yield, rate of decay and transport from the fuel matrix to the free volume. Upon changing the heat rating, a new equilibrium concentration will be reached, but not immediately. The time lag, which is specific to the isotope, is due both to the decay and to a delay in the fission gas release mechanism. To accurately calculate the available amount of short-lived iodine isotopes under a varying load, this time lag must be taken into account. Assuming c~i moles of the ith isotope inside the fuel and fli moles of the same isotope in the fuel rod void volume, the change in gas inventory can be written as ~ti = Y, V f f -- 2i o~i -- ~9, o~i

where f (fissions m - 3 s 1) is the fuel fission rate, V f (m 3) is the fuel volume, Yi (mol fission 1) is the molar fission yield of isotope i, 2i (s 1) is the decay constant of isotope i and ,% (s t) is the gas release time constant of isotope i. The fission yield and decay constants for five short-lived isotopes considered in the failure model are listed in Table 1. The gas release time constant 9~ is dependent on the diffusional transport of the newly produced isotope to the UO2 grain boundaries. An approximate expression for ~9~ can be found using a collection-of-spheres model for the gas diffusion (Hagrman, 1981): O/= 3 (DAi)1/2

(9)

a

where D (m e s 1) is the gas diffusion coefficient of UO2 and a (m) is the effective radius of gas diffusion. D and a are functions of fuel temperature and porosity. In the failure model, simple correlations are used for their determination. Solving Eq. (8) for each of the isotopes in Table 1 separately, the fluctuations in iodine inventory can be determined provided that the time-varying fission rate and fuel temperature are known. The iodine inventory is not uniformly distributed in the fuel rod void volume. In the failure model, released iodine is supposed to accumulate in areas close to pellet pellet axial interfaces and at pellet radial crack openings. The released iodine is thus assumed to affect only one-tenth of the total clad area when calculating the surface density needed in Eq. (5). The same reduced area is also considered in Eq. (4) when determining the probability of finding a surface defect of critical length in the fuel rod. Table 1 F i s s i o n yield a n d d e c a y c o n s t a n t s Isotope

F i s s i o n yield ( m o l fission - 1)

Decay constant (s - 1)

1311 1321 133I

5.15 7.14 1.15 1.31 1.01

9.97 8.37 9.26 2.20 2.87

134I

(8)

397

1351

X X X × ×

10 26 10 - 2 6 10 26 10 25 10 - 2 5

X × X x ×

10 - 7 10 - 5 10 6 10 4 10 - 5

398

L.O. Jernkvist / Nuclear Engineering and Design 156 (1995) 393 399

Neither the transport of iodine from the clad inner surface to the crack tip nor the chemical reactions deteriorating the cohesive strength of the material in front of the crack tip are explicitly treated by the failure model. These time-dependent phenomena, characteristic of environmentally assisted cracking, are instead implicitly accounted for by the correlation in Eq. (5).

5. Implementation of the failure model The presented failure model has been incorporated in the ABB A t o m STAV-T code, a two-dimensional finite element p r o g r a m m e specially developed to model fuel behaviour under both operational and off-normal transient conditions (Massih, 1993). Thermal and mechanical calculations can be performed in both the axisymmetric and the crosssectional geometry of the fuel rod. In the axisymmetric geometry either a full-length rod or a segment corresponding to the height of one or two pellets can be flexibly modelled by the finite element technique. Input to the code, other than the rod geometry, consists of prescribed initial conditions and the subsequent power history that is to be simulated. The desired power history is followed in an adaptive time-stepping scheme where thermal and mechanical calculations are performed at each of the time steps. The calculations are first performed in the axisymmetric geometry, whereupon a more detailed modelling is done in the cross-sectional p r o g r a m m e mode. In the finite element cross-sectional model, initial radial cracks of arbitrary length and symmetric spacing can be simulated by relaxing the boundary conditions for an appropriate number of nodes on the symmetry axis (Fig. 4). Propagation of cladding cracks is simulated by a node release technique. The finite element mesh near the propagating crack is locally refined, so that a nodal spacing of roughly 10 lam is acquired in the crack propagation direction. At any time step of the prescribed power history, incremental crack growth is modelled by relaxing a number of nodes in front of the crack tip. The number of nodes relaxed is imposed by the crack propagation

Fig. 4. Finite element mesh used in the cross-sectional geometry when calculating the mechanical interaction between a pellet and a cladding tube with four symmetrically spaced incipient cracks. velocity calculated through the correlations stated in Sections 3.1 and 3.2.

6, Application of the failure model; an example To illustrate the applicability of the failure model, the influence of pellet-clad friction on cladding crack formation has been investigated for an ABB A t o m P W R 17 x 17 X L fuel rod. Four symmetrically spaced incipient flaws, each 14.3 lam long, were assumed in the cladding. According to Eq. (4), the probability of finding a surface flaw of that length in the fuel rod cladding area considered to be exposed to iodine was 0.50. The flaws were modelled right opposite radial pellet cracks, inducing localized tangential straining in the cladding. The fuel rod properties are summarized in Table 2. The pellet-clad interaction was studied under the three-step transient load shown in Fig. 5 using Table 2 Properties of the modelled fuel rod Item

Value

Clad inner diameter Clad outer diameter Pellet outer diameter Rod average burn-up Clad initial flaw size Pellet crack size

8.36 mm 9.50 mm 8.19 mm 32.60 MWd kgU i 14.30 jam 2.05 mm

L.O. Jernkvist / Nuclear Engineering and Design 156 (1995) 393-399 7. Conclusions

65

/

6055"

/

50-

45'

T h e p r e s e n t e d P C I f a i l u r e m o d e l , u s i n g explicit m o d e l l i n g o f c l a d d i n g c r a c k g r o w t h in a finite element computational framework, allows the P C I f a i l u r e p r o c e s s t o be t r e a t e d as a l o c a l i z e d and time-dependent phenomenon. T h e m o d e l has p r o v e n p a r t i c u l a r l y useful w h e n s t u d y i n g the i n f l u e n c e o f t i m e - d e p e n d e n t p a r a m e ters, e.g. r a m p rates a n d m a t e r i a l creep, o n P C I i n d u c e d c l a d d i n g failures.

~_~ 4 0 "

30 25

ffi 150 W / m s

20

References

15 0

i

~

|

i

i

,

a

2

4

6

8

10

12

14

16

'Ifme [ rain ] Fig. 5. Transient load applied to the modelled fuel rod. t h e A B B A t o m STAV-T fuel p e r f o r m a n c e c o d e . B e g i n n i n g f r o m 19 k W m ~, a final h e a t r a t e o f 60 k W m ~ w a s r e a c h e d a f t e r r o u g h l y 12 m i n . T h e r a m p r a t e w a s fixed to 150 W m s - 1 . In Fig. 6 the c a l c u l a t e d c r a c k l e n g t h is p l o t t e d as a f u n c t i o n o f t i m e f o r t h r e e d i f f e r e n t v a l u e s o f t h e coefficient o f p e l l e t - c l a d f r i c t i o n . T h e h i g h e s t v a l u e o f / ~ results in c l a d d i n g failure, w h e r e a s f o r t h e l o w e r v a l u e s o f / t t h e m e c h a n i c a l i n t e r a c t i o n is n o t sufficient to i n d u c e t h r o u g h - w a l l c r a c k s e v e n t h o u g h t h e h e a t r a t i n g is 60 k W m -~. 240

~=1.50

2OO

16o

~

__~ffil.O0

( / ~ f f i O . 7 ~ 40 ¸

0 0

399

|

!

i

l

i

,

!

2

4

6

8

10

12

14

Time [rain] Fi~. 6. Calculated crack growth.

16

L. Brunisholz and C. Lemaignan, Iodine-induced stress corrosion of zircaloy fuel cladding: initiation and growth, ASTM STP 939 (1987) 700-716. B. Cox and R. Haddad, Recent studies of crack initiation during stress corrosion cracking of zirconium alloys, ASTM STP 939 (1987) 717 733. K. Forsberg and A.R. Massih, Fission gas release under time varying conditions, J. Nucl. Mater. 127 (1985) 141. D.L. Hagrman, Cesium and iodine release (CESIOD), MATPRO version 11; a handbook of materials properties for use in the analysis of light water reactor fuel rod behaviour, NUREG Rep. CR-0497, Revision 2, 1981. A.R. Massih, T. Rajala and L.O. Jernkvist, Analysis of pellet cladding mechanical interaction behavior of different ABB Atom fuel rod designs, Trans. 12th Int. Conf. on Structural Mechanics in Reactor Technology, Vol. C, 1993, pp. 57 68. A.K. Miller, K.D. Challenger and A. Tasooji, SCCIG: a phenomenological model for iodine stress corrosion cracking of zircaloy, EPRI Rep. NP-1798, 1981. M. Nagai, S. Shimada, S. Nishimura, H. Ueda and G. Yagawa, Evaluation of SCC crack behavior in zirconium and Zr-2 using non-linear fracture mechanics parameters, Nucl. Eng. Des. 88 (1985) 319-326. J.R. Rice, A path-independent integral and the approximate analysis of strain concentration by notches and cracks, J. Appl. Mech. 35 (1968) 379 386. W.S. Ryu, Y.H. Kang and Y.J. Lee, Effects of iodine concentration on iodine-induced SCC of zircaloy-4 tubes, J. Nucl. Mater. 152 (1988) 194-203. I. Schuster and C. Lemaignan, Influence of texture on iodineinduced SCC of zircaloy-4 cladding tubes, J. Nucl. Mater. 189 (1992) 157 166. I. Schuster, C. Lemaignan and J. Joseph, Influence of irradiation on iodine induced stress corrosion cracking behaviour of zircaloy 4, Trans. 12th Int. Conf. on Structural Mechanics in Reactor Technology, Vol. C, 1993, pp. 45 50. C.F. Shih, H.G. deLorenzi and W.R. Andrews, Studies on crack initiation and stable crack growth, ASTM STP 668 (1979) 65-120. W. Weibull, A statistical distribution function of wide applicability, J. Appl. Mech. 18 (1951) 293-297.